Warm-up
1. How many symmetry lines does a regular
hexagon have? How many centers of symmetry?
       6 symmetry lines, 1 c...
Section 3-4
Symmetries of Graphs
Reflection-symmetric: A figure that can be mapped
   onto itself by a reflection over a line




Axis/Line of symmetry: The l...
Symmetry to point P/point symmetry (180º-rotation-
symmetric: A figure that can be mapped onto itself
   under a 180º rotat...
Example 1
 Prove that the graph of y = x2 is symmetric to the
                       y-axis.
    We need to show that all ...
Symmetric to the origin:


A figure that is 180º-rotation-symmetric around the
                       origin
Example 2
Prove that the graph of y = x is symmetric to the
                    origin.
   We need to show that all points...
Power function: A function in the form y = xn,
    where n ≥ 2



Even function: A power function where n is
    even; x i...
Example 3
Does f(x) = 5x3 - 10x2 have line symmetry or point
          symmetry? How do you know?




             It has ...
The Graph-Translation Theorem
  helps us to figure out where
        asymptotes are.
Example 4
Give equations for the asymptotes for the following
                      graph:
                            3
 ...
Homework



p. 183 #1 - 21
Notes 3-4
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Notes 3-4

  1. 1. Warm-up 1. How many symmetry lines does a regular hexagon have? How many centers of symmetry? 6 symmetry lines, 1 center of symmetry 2. How many symmetry lines does an isosceles triangle have? How many centers of symmetry? 3 symmetry lines, 0 centers of symmetry 3. Draw a figure with more than one center of symmetry. Try y = sin(x) or y = x
  2. 2. Section 3-4 Symmetries of Graphs
  3. 3. Reflection-symmetric: A figure that can be mapped onto itself by a reflection over a line Axis/Line of symmetry: The line that a reflection- symmetric figure is reflected over
  4. 4. Symmetry to point P/point symmetry (180º-rotation- symmetric: A figure that can be mapped onto itself under a 180º rotation around point P Center of symmetry: The point P that a 180º- rotation-symmetric figure is rotated around Symmetric with respect to an axis: When a graph is a reflection over either the x-axis or y-axis
  5. 5. Example 1 Prove that the graph of y = x2 is symmetric to the y-axis. We need to show that all points map onto a corresponding coordinate where the x-coordinate is the opposite, and the y-coordinate is the same. Suppose (x, y) is on the graph. Then y = x2. Plug in -x for x and we get y = (-x)2. (-x)2 = x2. So when (x, y) is on the graph, so is (-x, y).
  6. 6. Symmetric to the origin: A figure that is 180º-rotation-symmetric around the origin
  7. 7. Example 2 Prove that the graph of y = x is symmetric to the origin. We need to show that all points map onto a corresponding coordinate where the x-coordinate and y-coordinate are opposite. Suppose (x, y) is on the graph. Then y = x, so -y = -x.
  8. 8. Power function: A function in the form y = xn, where n ≥ 2 Even function: A power function where n is even; x in the domain, f(-x) = f(x) Odd function: A power function where n is odd; x in the domain, f(-x) =- f(x)
  9. 9. Example 3 Does f(x) = 5x3 - 10x2 have line symmetry or point symmetry? How do you know? It has point symmetry. Can you find the point?  2 −80  ,   3 27 
  10. 10. The Graph-Translation Theorem helps us to figure out where asymptotes are.
  11. 11. Example 4 Give equations for the asymptotes for the following graph: 3 h( x ) = +4 x+2 This is a translated hyperbola. The parent function of a hyperbola has asymptotes € at x = 0 and y = 0. The translation is (x - 2, y + 4). x = -2 and y = 4
  12. 12. Homework p. 183 #1 - 21

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